Question

6. Suppose the production function is given by Q=1/20*L^1/2*K^1/4;price of labor(w) = 0.50 and price of capital (r) = 4: The market price for the output produced is P= 560: (a) Short-run production:

ii. Write down this firm's LONG-RUN cost minimization problem. [Note: In long-run, nothing is fixed, so the firm can choose both labor and capital optimally to minimize its cost.]

iii. Solving the long-run cost minimization problem, we get the long-run cost function C(Q) = (12)*(5Q)^4/3:

Find out the long-run cost of the firm for producing 30 units of output.

iv. Write down the functional forms of the long-run average cost and marginal cost.

v. Find out the profit maximizing choice of output and labor in the long-run. How much is the profit?

Answer 1

(ii)

Production finction f(k,l) is given below

we can write the cost function as TC = wL+rK

where w is cost per unit of labor and r isprice of capital per unit. We use the lagrange multiplier method to write down the firms long run minimization problem.

Z = wL+rK -u(Q-f(k,l))   Now, we miimize the Lagrangian with respect to two choice variable L and K and the lagrange multiplier u so,

Min (wL+rK -u(Q-f(k,l)) ).......................................................eq 1 with respect to L,K, and u

profit maximization (or cost minimization) he first-order conditions are:

partaial diffrentiation with respect to L in eq 1 (marginal prodction of labor)

partaial diffrentiation with respect to K in eq 1 (marginal prodction of capital)

from eq2 and eq3 we get

This condition is known as a Producer Optimum in the Long Run

If we put w = .5 r= 4 than when the above condition hold

(iii)

As we know C(Q) = wL+rK for cost minimization L=16K so

C(q) = w16K+rK put w= 0.5 and r =4

C(Q) = 8K+4K = 12K ..............................................eq 4 as we know that

Put L= 16K here also that is cost minimun codition we derived in the part (ii) so

so put this in eq 4 so

Proved.

Put Q= 30 in above equation

(iv)

Marginal Cost is   MC

(V) Profit maximization would happen when the cost is minimum we already know the minimum cost function C(Q)

Pr = PQ-C(Q) here C(Q) is minimum cost function than this would be maximum profit in the long run

Put the values to get the Maximum Profit.

P =560 , Profit will be maximum if

So When we DO this P =Marginal Cost

is the condition for profit maximization so

is Quantitiy for profit maximization